Lamé analytical solution for a hollow cylinder under radial pressure with a linear isotropic Hooke material. More...

#include "tutorials/src/AnalyticalSolutions.hpp"

Collaboration diagram for AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >:

Public Member Functions
	HollowCylinderUnderRadialPressure (const double inner_radius, const double outer_radius, const double inner_pressure, const double outer_pressure, const double young_modulus, const double poisson_ratio)

MoFEM::MatrixDouble	stress (const double x, const double y, const double) const
	Return the Cauchy stress in symmetric tensor storage at point (x, y, z).

MoFEM::VectorDouble	displacement (const double x, const double y, const double) const
	Return the displacement vector at point (x, y, z).

Static Private Member Functions
static constexpr double	sqr (const double v)

Private Attributes
const double	E

const double	nu

double	A

double	B

Detailed Description

template<int SPACE_DIM>
struct AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >

Lamé analytical solution for a hollow cylinder under radial pressure with a linear isotropic Hooke material.

The cylinder axis is aligned with z, while the solution is evaluated in the x-y cross-section. The z coordinate is ignored. SPACE_DIM == 2 is the plane-strain case with only the in-plane stress and displacement components returned. For SPACE_DIM == 3 this uses the plane-strain axial stress and zero axial displacement.

Examples: mofem/tutorials/vec-11_elastic_tie_mesh/elastic_tie_mesh.cpp.

Definition at line 18 of file AnalyticalSolutions.hpp.

Constructor & Destructor Documentation

◆ HollowCylinderUnderRadialPressure()

template<int SPACE_DIM>

AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >::HollowCylinderUnderRadialPressure	(	const double	inner_radius,
		const double	outer_radius,
		const double	inner_pressure,
		const double	outer_pressure,
		const double	young_modulus,
		const double	poisson_ratio
	)

inline

Definition at line 20 of file AnalyticalSolutions.hpp.

      : E(young_modulus), nu(poisson_ratio) {
    const double a2 = sqr(inner_radius);
    const double b2 = sqr(outer_radius);
    const double denom = b2 - a2;
 
    A = (inner_pressure * a2 - outer_pressure * b2) / denom;
    B = (inner_pressure - outer_pressure) * a2 * b2 / denom;
  }

Member Function Documentation

◆ displacement()

template<int SPACE_DIM>

MoFEM::VectorDouble AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >::displacement	(	const double	x,
		const double	y,
		const double
	)		const

inline

Return the displacement vector at point (x, y, z).

Definition at line 64 of file AnalyticalSolutions.hpp.

                                                       {
    MoFEM::VectorDouble disp(SPACE_DIM);
    disp.clear();
 
    const double r2 = sqr(x) + sqr(y);
    const double r = std::sqrt(r2);
    const double u_r =
        (1. + nu) / E * ((1. - 2. * nu) * A * r + B / r);
 
    disp[0] = u_r * x / r;
    disp[1] = u_r * y / r;
    if constexpr (SPACE_DIM == 3)
      disp[2] = 0.;
 
    return disp;
  }

◆ sqr()

template<int SPACE_DIM>

static constexpr double AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >::sqr ( const double v )

inlinestaticconstexprprivate

Definition at line 83 of file AnalyticalSolutions.hpp.

83{ return v * v; }

v

const double v

phase velocity of light in medium (cm/ns)

Definition initial_diffusion.cpp:41

◆ stress()

template<int SPACE_DIM>

MoFEM::MatrixDouble AnalyticalSolutions::HollowCylinderUnderRadialPressure< SPACE_DIM >::stress	(	const double	x,
		const double	y,
		const double
	)		const

inline

Return the Cauchy stress in symmetric tensor storage at point (x, y, z).

Definition at line 37 of file AnalyticalSolutions.hpp.

                                                 {
    MoFEM::MatrixDouble stress((SPACE_DIM * (SPACE_DIM + 1)) / 2, 1);
    stress.clear();
 
    const double r2 = sqr(x) + sqr(y);
    const double sigma_r = A - B / r2;
    const double sigma_t = A + B / r2;
    const double inv_r = 1. / std::sqrt(r2);
    const double c = x * inv_r;
    const double s = y * inv_r;
 
    auto t_stress = MoFEM::getFTensor2SymmetricFromMat<
        SPACE_DIM, -1,
        MoFEM::DataLayoutTraits<MoFEM::DataLayout::CoeffsByGauss>>(stress);
    t_stress(0, 0) = sigma_r * sqr(c) + sigma_t * sqr(s);
    t_stress(1, 1) = sigma_r * sqr(s) + sigma_t * sqr(c);
    t_stress(0, 1) = (sigma_r - sigma_t) * s * c;
    if constexpr (SPACE_DIM == 3)
      t_stress(2, 2) = 2. * nu * A;
 
    return stress;
  }